×

Large-time behavior of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity. (English) Zbl 0981.74011

The authors study the large-time behavior of smooth solutions to the equations of one-dimensional nonlinear thermoviscoelasticity. The constitutive equations considered in the paper have the form \(e=C_V\theta\), \(\sigma=-f(u)\theta +\hat\mu (\theta)v_x\) and \(q=-k\theta_x/u\), where \(C_V, k\) are positive constants, \(e\), \(\sigma\), \(q\) and \(\theta\), \(u\) and \(v\) are internal energy, stress, heat flux, and absolute temperature, specific volume and velocity, respectively; \(f(u)\) is a smooth function satisfying inequality \(f(u)\geq 0\), \(0<u<\tilde u\) and \(f(u)\leq 0\), \(\tilde U <u<\infty\) for some positive fixed \(\tilde u\leq \tilde U\); the viscosity \(\hat\mu (u)\) satisfies inequalities \(\hat\mu (u)u\geq \mu_0>0\) for \(u>0\). An initial-boundary value problem with stress-free and thermally insulated boundary conditions is studied. Using careful energy estimates and the techniques of G. Andrews and J. M. Ball [J. Differ. Equations 44, 306-341 (1982; Zbl 0501.35011)], the authors prove that as \(t\to\infty\), the pressure \(f(u)\theta\) respectively the velocity \(v\) tends to zero in the \(L^1\)- respectively \(L^2\)-norm, and \(\int^1_0\theta (x,t)dx\) tends to a constant. Moreover, there is a Young measure \(\nu_x\) on \(I R\) with supp\(\nu_x\subset\{z: f(z)=0\}\), such that if \(\Phi\in C(I R)\), then \(\Phi (u(\cdot, t))\rightharpoonup\langle\nu_x,\Phi\rangle\) weak-\(*\) in \(L^\infty (0,1)\) as \(t\to\infty\). This paper extends the above G. Andrews and J. M. Ball’s phase transition results to the nonisothermal case.

MSC:

74F05 Thermal effects in solid mechanics
74D10 Nonlinear constitutive equations for materials with memory
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics

Citations:

Zbl 0501.35011
PDFBibTeX XMLCite
Full Text: DOI